Unit 4 Seminar GRAPHS 4.1 Variation 4.2 Linear Inequalities 4.3 Graphing Linear Equations 1 1 1 1

4.1 Variation 2 2 2 2 2

Direct Variation If a variable y varies directly with a variable x, then y = kx, where k is the constant of proportionality. If x increases, then y increases. If x decreases, then y decreases. 3 3 3 3 3

The amount of interest earned is $70. EXAMPLE: The amount of interest earned on an investment, I, varies directly as the interest rate, r. If the interest earned is $50 when the interest rate is 5%, find the amount of interest earned when the interest rate is 7%. I = kr Write the variation equation, I varies directly r $50 = 0.05k Find k by substituting the given variables 1000 = k k = 1000, r = 7% I = 1000r Use k to determine I when r is given I = 1000(0.07) I = 70 The amount of interest earned is $70. 4 4 4 4 4

Inverse Variation If x increases, then y decreases. If a variable y varies inversely with a variable x, then , where k is the constant of proportionality. If x increases, then y decreases. If x decreases, then y increases. 5 5 5 5 5

Write the variation equation, y varies inversely as x EXAMPLE: Suppose y varies inversely as x. If y = 12 when x = 18, find y when x = 21. Write the variation equation, y varies inversely as x Find k by substituting the given variables Use k to determine y when x is given 6 6 6 6 6

Joint Variation The general form of a joint variation, where y varies directly as x and z, is y = kxz, where k is the constant of proportionality. As x and z increase, y increases. As x and z decrease, y decreases. 7 7 7 7 7

EXAMPLE: The area, A, of a triangle varies jointly as its base, b, and height, h. If the area of a triangle is 48 in2 when its base is 12 in. and its height is 8 in., find the area of a triangle whose base is 15 in. and whose height is 20 in. A varies jointly as b and h Using k, find A when b and h are given. Find k using the given values 8 8 8 8 8

Find the constant of proportionality. Now find A. EXAMPLE: A varies jointly as B and C and inversely as the square of D. If A = 1 when B = 9, C = 4, and D = 6, find A when B = 8, C = 12, and D = 5. Find the constant of proportionality. Now find A. 9 9 9 9 9

EVERYONE: pg 151 #44 Making an Ice Cube The time, t, for an ice cube to melt is inversely proportional to the temperature, T, of the water in which the ice cube is placed. If it takes an ice cube 2 minutes to melt in 75 degrees F water, how long will it take an ice cube of the same size to melt in 80 degrees F of water? 10 10 10 10

t = k/T 2 = k/75 150 = k t = k/T t = 150/80 t = 1.875 min ANSWER to: pg 151 #44 t = k/T 2 = k/75 150 = k t = k/T t = 150/80 t = 1.875 min 11 11 11 11 11

4.2 Linear Inequalities 12 12 12 12 12

Inequality signs < is less than (example: 3 < 5) ≤ is less than or equal to > is greater than (example: 5 > 3) ≥ is greater than or equal to 13 13 13 13 13

Graphing Inequalities x < 2 x ≤ 4 The numbers less than to 2 are all the points on the number line to the left of 2. The open circle at 2 shows that 2 is NOT included in the solution set. The numbers less than or equal to 4 are all the points on the number line to the left of 4 and 4 itself. The closed circle at 4 shows that 4 is included in the solution set. 14 14 14 14 14

Graphing Inequalities x > 3 x ≥ -5 The numbers greater than to 3 are all the points on the number line to the right of 3. The open circle at 3 shows that 3 is NOT included in the solution set. The numbers greater than or equal to -5 are all the points on the number line to the right of -5 and -5 itself. The closed circle at -5 shows that -5 is included in the solution set. 15 15 15 15 15

Example: Solve and graph the solution Solve 3x – 8 < 10 and graph the solution set. The solution set is all real numbers less than 6.

When dividing by a negative number, flip the inequality sign. 10 - 3x ≤ 21 10 - 3x ≤ 21 -3x ≤ 11 When dividing by a negative number, flip the inequality sign. 17 17 17 17 17

When multiplying by a negative number, flip the inequality sign. 18 18 18 18 18

Compound Inequality Graph the solution set of the inequality 4 < x  3 where x is a real number The solution set consists of all real numbers between 4 and 3, including the 3 but not the 4.

Get the x alone in the middle… 20 20 20 20 20

Example A student must have an average (the mean) on five tests that is greater than or equal to 85% but less than 92% to receive a final grade of B. Jamal’s scores on the first four tests were 98%, 89%, 88%, and 93%. What range of scores on the fifth test will give him a B in the course?

Example (continued) Let x = Jamal’s score on the fifth test. Then: So Jamal will receive a grade of B in the course if his score on the fifth test is greater than or equal to 57 and less than 92.

4.3 Graphing Linear Equations 23 23 23 23 23

Rectangular Coordinate System The rectangular coordinate system consists of a horizontal axis called the x-axis and a vertical axis called the y-axis. The x- and y-axes intersect at a point called the origin, dividing the plane into four regions called quadrants. 24 24 24 24

The x and y values are called the coordinates of the point. Each point in the plane corresponds to an ordered pair (x, y) of real numbers. The x and y values are called the coordinates of the point. The coordinates for the origin are (0, 0). The x-coordinate tells how many units the point is to the left or right of the origin. If x is positive, the point is to the right of (0, 0). If x is negative, the point is to the left of (0, 0). The y-coordinate tells how many units the point is up or down from the origin. If y is positive, the point is above (0, 0). If y is negative, the point is below (0, 0). Locating a point in the plane is called plotting the point. 25 25 25 25

Example: Plot the point (3, 5). Move right along the horizontal x-axis to 3. Move up 5 units until aligned with the 5 on the vertical y-axis. Draw a point (3, 5) 26 26 26 26

Example: Plot the point (4, –2). Move right along the horizontal x-axis to 4. Move down 2 units until aligned with the –2 on the vertical y-axis. (4, –2) 27 27 27 27

Example: Plot the point (–2, 4). Move left along the horizontal x-axis to –2. Move up 4 units until aligned with the 4 on the vertical y-axis. (–2, 4) 28 28 28 28

Example: Plot the point (–3, 0). Move left along the horizontal x-axis to –3. The 0 indicates that we do not move along the vertical y-axis. The point is plotted on the x-axis. (–3, 0) 29 29 29 29

Example: Plot the point (0, –5). The 0 indicates that we do not move along the horizontal x-axis. Move down 5 units to the –5 on the vertical y-axis. The point is plotted on the y-axis. (0, –5) 30 30 30 30

Linear Equations Linear equations are a type of equation whose graph is a straight line. The solutions of a linear equation are the ordered pairs (x, y) that make the equation true. The solutions are points that are on the line. 31 31 31 31

Example: Use a table of values to sketch the graph of the linear equation 2x - y = 4. Solve the equation for y. 2x - y = 4 is equivalent to y = 2x – 4 Select at least 3 values for x and create a table of values. 32 32 32 32

Example: cont’d Plot the points from the table and connect them using a straight line with arrow tips on the ends. 33 33 33 33

Intercepts The x-intercept of a linear equation graph is the point where the line crosses the x-axis. An x-intercept has the form (x, 0). The y-intercept of a linear equation graph is the point where the line crosses the y- axis. A y-intercept has the form (0, y). 34 34 34 34

The x-intercept of this graph is (2, 0). The y-intercept of this graph is (0, -4). x-intercept (2,0) y-intercept (0,-4) 35 35 35 35

Example: Draw the graph of the linear equation y = 3x + 6 using the x- and y-intercepts. The x-intercept is found by letting y = 0 and solving for x. y = 2x + 4 0 = 2x + 4 4 = 2x 2 = x The y-intercept is found by letting x = 0 and solving for y. y = 2(0) + 4 y = 4

Example: Graph 3x + 2y = 6 Find the x-intercept. 3x + 2y = 6 Find the y-intercept. 3(0) + 2y = 6 2y = 6 y = 3

Slope The slope of a line can be defined as the steepness of the line. If you want to find the slope of a straight line, choose any two points on the line and count the number of units in the rise (up or down) and the run (right) from one point to the other. The slope is the ratio of the rise to the run. 38 38 38 38

Slope, cont’d For example, you can determine the slope of an airplane’s flight path shortly after takeoff, as shown below. 39 39 39 39

For  , you can "rise" up or down but, you ALWAYS "run" to the right. 

Example: Calculate the slope of the line. 41 41 41 41

Example: Solution From point A to point B we count up 4 units and run 7 units. The slope is Notice that the slope goes uphill and  is a positive number 42 42 42 42

Example: Calculate the slope of the line. 43 43 43 43

Slope, cont’d Lines with positive slopes are increasing as they move from left to right. Lines with negative slopes are decreasing as they move from left to right. 44 44 44 44

Slope, cont’d We can also calculate the slope of a line, without looking at the graph, if we know two points on the line. 45 45 45 45

Example: Calculate the slope of the line. 46 46 46 46

Example: Calculate the slope of the line that passes through the points (0, -5) and (1, -3). Solution: Using (0, -5) and (1, -3), The slope of the line is 2. 47 47 47 47

Example: Calculate the slope of the line that passes through the points (2, 3) and (2, -1). Solution: : Using (2, 3) and (2, -1), The slope of the line is undefined. . 48 48 48 48

Example: cont’d Solution, cont’d: If the slope of a line is undefined, the graph is a vertical line. 49 49 49 49

Example: Calculate the slope of the line that passes through the points (-2, 4) and (3, 4). Solution: Using (-2, 4) and (3, 4), The slope of the line is 0. 50 50 50 50

Example: cont’d Solution, cont’d: If the slope of a line is 0, the graph is a horizontal line. 51 51 51 51

Summary: Types of Slope Positive slope rises from left to right. Negative slope falls from left to right. The slope of a vertical line is undefined. The slope of a horizontal line is zero. zero negative undefined positive

Graphing Linear Equations Using Slope Intercept Form If the general form of a linear equation is solved for y, the equation is then in slope-intercept form. 53 53 53 53

Example: It crosses the y-axis at 4, the slope is so we so we start there: the slope is     so we

Example: It crosses the y-axis at -2, the slope is 4 which is really so we start there: the slope is 4 which is really

Horizontal Lines A horizontal line has an equation of the form y = b. The equation of the graph shown is y = 3 The slope of a horizontal line is 0! y = 3 56 56 56 56

Vertical Lines A vertical line has an equation of the form x = a. The equation of the graph shown is x = 2 The slope of a vertical line is undefined! x = 2 57 57 57 57